PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

In this article,two results concerning the periodic points and normality of meromorphic functions are obtained:(i)the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by let ting R(z)be a nonpolynomial rational function,and if all zeros and poles of R(z)—z are multiple,then Rk(z)has at least k+1 fixed points in the complex plane for each integer k≥2;(ii)a complete solution to the problem of normality of meromorphic functions with periodic points is given by let ting F be a family of meromorphic functions in a domain D,and let ting k≥2 be a positive integer.If,for each f∈F,all zeros and poles of f(z)-z are multiple,and its iteration fk has at most k distinct fixed points in D,then F is normal in D.Examples show that all of the conditions are the best possible.

Least Number of Periodic Points of Self-maps of Lie Groups

There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M → M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g is continuous} and NJDn（f） = min{#Fix（gn）; g - f; g is smooth}. In general, NJDn（f） may be much greater than NFn（f）. If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds 〈=〉 all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.

Self-maps of S2 Homotopic to a Smooth Map with a Single n-periodic Point

We show for which （d, n） ∈ Z × N there exists a smooth self-map f ： S2 → S2 so that deg（f） = d and Fix（fn） is a point.

A Note on Quasi-weakly Almost Periodic Point

Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597- 606 （2013）] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy： A brief survey of recent results. Nonlinearity, 17, 493-502 （2004）]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.

FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we introduce new concepts of a-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21, 22] and different from a-GF-contractions given in [8]. Then, sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings, in the setting of complete metric space. Consequently, the obtained results encompass various generalizations of the Banach contraction principle. Moreover, some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

ON THE EXISTENCE OF POSITIVE SOLUTIONS TO SECOND ORDER PERIODIC BVPS

The existence of positive solutions to second-order periodic BVPs-u'+Mu =j(t, u),t(0) = u(2π),u'(0) = '(2π) and u'+ Mu = I(t, u), u(0) = u(2π), u'(0) = u'(2π)is proved by a simple appliCation of a Fixed point Theorem in cones due to Krasnoselskii.

ON PROPERTIES OF HYPERCHAOS: CASE STUDY

Some properties of hyperchaos are exploited by studying both uncoupled and coupled CML. In addition to usual properties of chaotic strange attractors, there are other interesting properties, such as: the number of unstable periodic points embedded in the strange attractor increases dramatically increasing and a large number of low-dimensional chaotic invariant sets are contained in the strange attractor. These properties may be useful for regarding the edge of chaos as the origin of complexity of dynamical systems.

A Note on “Multi-separation, Centrifugality and Centripetality Imply Chaos”

In a recent paper from Trans. Amer. Math. Soc., 351(1)(1999), 343—351, the maps f and g from [0,n] to itself are constructed, which have periodic points of period 2a+3 . In this note we prove that they have no periodic points of period 2a+1.

TOPOLOGICAL ENTROPY AND IRREGULAR RECURRENCE

This paper is devoted to problems stated by Z. Zhou and ELi in 2009. They concern relations between almost periodic, weakly almost periodic, and quasi-weakly almost periodic points of a continuous map f and its topological entropy. The negative answer follows by our recent paper. But for continuous maps of the interval and other more general one-dimensional spaces we give more results; in some cases the answer is positive.

ω-Limit Set of a Tree Map

Let T be a tree and f be a continuous map from T into itself. We show mainly in this paper that a point x of T is an w-limit point of f if and only if every open neighborhood of x in T contains at least nx + 1 points of some trajectory, where nx equals the number of connected components of T \ {x}. Then, for any open subset G w(f) in T, there exists a positive integer m = m(G) such that at most m points of any trajectory lie outside G.This result is a generalization of the related result for maps of the interval.

AN IMPROVEMENT AND PROOF OF OGY METHOD

OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and gives a mathematics proof of it.

A New Critical Value Concerning the Genealogy of Long Period Families at L_4 in the Restricted Three-body Problem

We found another critical mass ratio value -↑μ between μ4 and μ5 concerning the genealogy of the long period family around the equilateral equilibrium point L4 in the restricted three-body problem. This value has not been pointed out before. We used numerical computations to show how the long period family evolves around this critical value. The case is similar to that of the critical values between μ2 and μ4, with slight difference in evolution details.

FIXED POINTS AND EXPONENTIAL STABILITY OF ALMOST PERIODIC MILD SOLUTIONS TO STOCHASTIC VOLTERRA-LEVIN EQUATIONS

In this paper, we consider stochastic Volterra-Levin equations. Based on semigroup of operators and fixed point method, under some suitable assumptions to ensure the existence and stability of pth-mean almost periodic mild solutions to the system.

Extended tanh-function Method for Solving Traveling Wave Solutions of Nonlinear Kundu Equation

In this paper, we study the existence of multiple positive periodic solutions for the second order differential equation x′′（t） ＋ p（t）x′（t） ＋ q（t）x（t） = f（t, x（t））.By using Krasnoselskii fixed point theorem, we establish some criteria for the existence and multiple positive periodic solutions for this differential equation.

Equicontinuity of Maps on a Dendrite with Finite Branch Points

Let （T, d） be a dendrite with finite branch points and f be a continuous map from T to T. Denote by w（x, f） and P（f） the w-limit set of x under f and the set of periodic points of f, respectively. Write Ω（x, f） = {yl there exist a sequence of points xk ∈ T and a sequence of positive integers n1 〈 n2 〈 … such that lim k→∞ Xk = X and lim k→∞ f nk （xk） = y}. In this paper, we show that the following statements are equivalent： （1） f is equicontinuous. （2） w（x, f） = Ω（x, f） for any x ∈ T. （3） ∩ ∞ n=1 f n（T） = P（f）, and w（x, f） is a periodic orbit for every x ∈ T and map h： x → w（x, f） （x ∈ T） is continuous. （4） Ω（x, f） is a periodic orbit for any x ∈ T.

Banach Upper Density Recurrent Points of C^0-flows

Let X denote a compact metric space with distance d and F ： X×R→ X or Ft ： X→X denote a C0-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C0-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.

Kneading Sequences for Unimodal Expanding Maps of the Interval

Let P and AC be two primary sequences with min{P, AC} RLR∞,ρ（P） and p（AC） be the eigenvalues of P and AC, respectively. Let f ∈ C0（I, I） be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved that f has the kneading sequence K（f） （RC）*m * P if λ （ρ（P））1/2m, and K（f）】（RC）*m * AC * E for any shift maximal sequence E if λ】（ρ（AC））1/2m. The value of （ρ（P））1/2m or （ρ（AC））1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one.

An Arc as the Inverse Limit of a Single Bonding Map of Type N on [0,1]

Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_1=max(A\{a}),and B={x∈[c_2,1]|f(x)=x},b=minB,b_1=min(B\{b}).Then the in- verse limit(I,f)is an arc if and only if one of the following three conditions holds: (1)If c_1<f(c_1)≤c_2(resp.c_1≤f(c_2)<c_2),then f has a single fixed point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)<b). (2)If f(c_1)≤c_1(resp.f(c_2)≥c_2),then f has more than one fixed point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1<f(c_2)<a,f^2(c_2)>f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if b<f(c_2)<b_1,f^2(c_1)<f(c_1)). (3)If f(c_1)>c_2 and f(c_2)<c_1,then f has a single fixed point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.

Non-Wandering Sets of the Powers of Dendrite Maps

Let （X, d） be a metric space and f be a continuous map from X to X. Denote by EP（f） and Ω（f） the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold： （1） If x ∈ Ω（f） - Ω（f^n） for some n ≥ 2, then x ∈ EP（f）. （2） Ω（f） is contained in the closure of EP（f）. The aim of this note is to show that the above results do not hold for maps of dendrites D with Card（End（D）） = No （the cardinal number of the set of positive integers）.

Fast approximate maximum likelihood period estimation from incomplete timing data

To estimate the period of a periodic point process from noisy and incomplete observations, the classical periodogram algorithm is modified. The original periodogram algorithm yields an estimate by performing grid search of the peak of a spectrum, which is equivalent to the periodogram of the periodic point process, thus its performance is found to be sensitive to the chosen grid spacing. This paper derives a novel grid spacing formula, after finding a lower bound of the width of the spectral mainlobe. By employing this formula, the proposed new estimator can determine an appropriate grid spacing adaptively, and is able to yield approximate maximum likelihood estimate （MLE） with a computational complexity of O（n2）. Experimental results prove that the proposed estimator can achieve better trade-off between statistical accuracy and complexity, as compared to existing methods. Simulations also show that the derived grid spacing formula is also applicable to other estimators that operate similarly by grid search.